The distribution of $X$ is the distribution of the ratio 
$$\frac{\sum_{i=1}^n\lambda_iZ_i^2}{\sum_{i=1}^nZ_i^2}
$$
of two quadratic forms in iid standard normal random variables $Z_1,\dots,Z_n$ (because the distribution of $(v_1,\dots,v_n)$ is the same as that of $(Z_1,\dots,Z_n)\big/\sqrt{\sum_{i=1}^nZ_i^2}$). The distribution of such ratios was studied by [Gurland][1]; also see e.g. [Watson][2]. 


  [1]: https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=3&ved=2ahUKEwiDoeDbrP3hAhUOEawKHTKNBtsQFjACegQIAxAC&url=https%3A%2F%2Fcowles.yale.edu%2Fsites%2Fdefault%2Ffiles%2Ffiles%2Fpub%2Fcdp%2Fs-0354.pdf&usg=AOvVaw1xMd5R-PTGs0rvjHtme9W8
  [2]: http://adsabs.harvard.edu/full/1955AuJPh...8..402W